Triangle calculator ASA

Please enter the side of the triangle and two adjacent angles
°
°


Right scalene triangle.

Sides: a = 0.28880177225   b = 0.85502671395   c = 0.8

Area: T = 0.1155207089
Perimeter: p = 1.93882848619
Semiperimeter: s = 0.9699142431

Angle ∠ A = α = 19.8° = 19°48' = 0.34655751919 rad
Angle ∠ B = β = 90° = 1.57107963268 rad
Angle ∠ C = γ = 70.2° = 70°12' = 1.22552211349 rad

Height: ha = 0.8
Height: hb = 0.27109903362
Height: hc = 0.28880177225

Median: ma = 0.81328582608
Median: mb = 0.42551335697
Median: mc = 0.49329038532

Inradius: r = 0.11988752915
Circumradius: R = 0.42551335697

Vertex coordinates: A[0.8; 0] B[0; 0] C[0; 0.28880177225]
Centroid: CG[0.26766666667; 0.09660059075]
Coordinates of the circumscribed circle: U[0.4; 0.14440088612]
Coordinates of the inscribed circle: I[0.11988752915; 0.11988752915]

Exterior(or external, outer) angles of the triangle:
∠ A' = α' = 160.2° = 160°12' = 0.34655751919 rad
∠ B' = β' = 90° = 1.57107963268 rad
∠ C' = γ' = 109.8° = 109°48' = 1.22552211349 rad

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How did we calculate this triangle?

1. Calculate the third unknown inner angle

 alpha = 19° 48' ; ; beta = 90° ; ; ; ; alpha + beta + gamma = 180° ; ; ; ; gamma = 180° - alpha - beta = 180° - 19° 48' - 90° = 70° 12' ; ;

2. By using the law of sines, we calculate unknown side a

c = 0.8 ; ; ; ; fraction{ a }{ c } = fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = c * fraction{ sin( alpha ) }{ sin ( gamma ) } ; ; ; ; a = 0.8 * fraction{ sin(19° 48') }{ sin (70° 12') } = 0.29 ; ;

3. By using the law of sines, we calculate last unknown side b

 fraction{ b }{ c } = fraction{ sin( beta ) }{ sin ( gamma ) } ; ; ; ; b = c * fraction{ sin( beta ) }{ sin ( gamma ) } ; ; ; ; b = 0.8 * fraction{ sin(90° ) }{ sin (70° 12') } = 0.85 ; ;


Now we know the lengths of all three sides of the triangle and the triangle is uniquely determined. Next we calculate another its characteristics - same procedure as calculation of the triangle from the known three sides SSS.

a = 0.29 ; ; b = 0.85 ; ; c = 0.8 ; ;

4. The triangle circumference is the sum of the lengths of its three sides

p = a+b+c = 0.29+0.85+0.8 = 1.94 ; ;

5. Semiperimeter of the triangle

s = fraction{ o }{ 2 } = fraction{ 1.94 }{ 2 } = 0.97 ; ;

6. The triangle area using Heron's formula

T = sqrt{ s(s-a)(s-b)(s-c) } ; ; T = sqrt{ 0.97 * (0.97-0.29)(0.97-0.85)(0.97-0.8) } ; ; T = sqrt{ 0.01 } = 0.12 ; ;

7. Calculate the heights of the triangle from its area.

T = fraction{ a h _a }{ 2 } ; ; h _a = fraction{ 2 T }{ a } = fraction{ 2 * 0.12 }{ 0.29 } = 0.8 ; ; h _b = fraction{ 2 T }{ b } = fraction{ 2 * 0.12 }{ 0.85 } = 0.27 ; ; h _c = fraction{ 2 T }{ c } = fraction{ 2 * 0.12 }{ 0.8 } = 0.29 ; ;

8. Calculation of the inner angles of the triangle using a Law of Cosines

a**2 = b**2+c**2 - 2bc cos( alpha ) ; ; alpha = arccos( fraction{ a**2-b**2-c**2 }{ 2bc } ) = arccos( fraction{ 0.29**2-0.85**2-0.8**2 }{ 2 * 0.85 * 0.8 } ) = 19° 48' ; ; beta = arccos( fraction{ b**2-a**2-c**2 }{ 2ac } ) = arccos( fraction{ 0.85**2-0.29**2-0.8**2 }{ 2 * 0.29 * 0.8 } ) = 90° ; ; gamma = arccos( fraction{ c**2-a**2-b**2 }{ 2ba } ) = arccos( fraction{ 0.8**2-0.29**2-0.85**2 }{ 2 * 0.85 * 0.29 } ) = 70° 12' ; ;

9. Inradius

T = rs ; ; r = fraction{ T }{ s } = fraction{ 0.12 }{ 0.97 } = 0.12 ; ;

10. Circumradius

R = fraction{ a }{ 2 * sin( alpha ) } = fraction{ 0.29 }{ 2 * sin 19° 48' } = 0.43 ; ;

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